7.3.9. Quantum Tunneling

Table of Contents

A Barrier Is Not Always a Barrier

In classical physics, a particle can cross a barrier only if it has enough energy. If a ball rolls toward a hill, it must have at least as much energy as the height of the hill to get over it. In quantum mechanics, the situation is different. A particle is described by a wave function, and this wave can extend into regions that would be forbidden in classical physics. Because of this, there is a nonzero chance that the particle appears on the other side of the barrier. This phenomenon is called quantum tunneling.

Tunneling is one of the clearest examples of how quantum behavior differs from everyday experience. It does not mean the particle digs a hole through the barrier. It means that the wave function penetrates the barrier and may still have a finite value beyond it, giving a probability of finding the particle there.

A particle with energy $E$ less than a barrier height $V_0$ can still pass through the barrier in quantum mechanics.
Classically, if $E < V_0$, crossing is impossible.
Quantum mechanically, if $E < V_0$, transmission can still occur with some probability.

A Simple Barrier Model

To understand tunneling, consider a one dimensional potential barrier. The potential energy is zero on the left, rises to a constant value $V_0$ over some finite width $a$, and then returns to zero on the right.

$$
V(x) =
\begin{cases}
0, & x < 0 \\
V_0, & 0 \le x \le a \\
0, & x > a
\end{cases}
$$

Suppose a particle approaches from the left with energy $E$, where $E < V_0$. In classical mechanics, the particle would reflect back. In quantum mechanics, the wave function enters the barrier and decreases there, but it usually does not become exactly zero. If some of the wave reaches the far side, there is a chance the particle is transmitted.

Rendering scientific sketch…

Finite potential barrier and tunneling

What the Wave Function Does Inside the Barrier

The time independent Schrödinger equation is

$$
-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi
$$

Inside the barrier, where $V(x)=V_0$ and $E<V_0$, this becomes

$$
-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V_0 \psi = E\psi
$$

which can be rearranged to

$$
\frac{d^2\psi}{dx^2} = \kappa^2 \psi
$$

where

$$
\kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}
$$

The solutions are exponential rather than oscillating:

$$
\psi(x) \propto e^{\kappa x} \quad \text{or} \quad e^{-\kappa x}
$$

This is the key mathematical feature of tunneling. In a classically forbidden region, the wave function does not oscillate like a traveling wave. Instead, it decays exponentially.

For a region where $E < V(x)$, the wave function typically decreases exponentially.
A common decay factor is
$$
\psi(x) \sim e^{-\kappa x}, \qquad \kappa = \frac{\sqrt{2m(V - E)}}{\hbar}
$$
Larger barrier height, larger barrier width, or larger particle mass makes tunneling less likely.

Transmission Probability

The main physical question is, what fraction of particles tunnel through the barrier? This is described by the transmission probability, usually written as $T$.

For a sufficiently thick or high barrier, an important approximate result is

$$
T \propto e^{-2\kappa a}
$$

where $a$ is the barrier width. This formula shows the essential behavior. The transmission probability drops exponentially when the barrier gets wider or higher.

This means tunneling can be very sensitive to small changes in distance. Even a tiny increase in barrier width can strongly reduce the chance of transmission.

Barrier property	Effect on tunneling
Larger width $a$	Less tunneling
Larger height $V_0$	Less tunneling
Larger mass $m$	Less tunneling
Energy closer to $V_0$	More tunneling

A useful tunneling estimate is
$$
T \propto e^{-2\kappa a}
$$
This exponential dependence is one of the most important results in quantum tunneling.

Why Tunneling Does Not Violate Energy Conservation

At first, tunneling can seem strange. If the particle crosses a barrier higher than its energy, does it somehow borrow extra energy? In standard quantum mechanics, the answer is no. Energy is still conserved.

The particle has total energy $E$ before and after the barrier. The barrier does not give the particle extra energy. Instead, the wave nature of the particle allows a nonzero probability of appearing beyond the barrier. The unusual part is the probability behavior, not a failure of conservation laws.

Tunneling Through Different Kinds of Barriers

The finite rectangular barrier is the simplest example, but tunneling is much more general. If the potential changes smoothly rather than sharply, tunneling can still occur whenever a particle encounters a region where $E < V(x)$.

In such cases, the wave function usually decays through the forbidden region and may reappear beyond it. The exact transmission probability depends on the detailed shape of the potential, but the same basic idea remains true.

Physical Meaning

Quantum tunneling shows that particles are not tiny classical balls following only one precise path. Their behavior is governed by the wave function, and the wave function can extend into forbidden regions. Because measurement probabilities come from the wave function, a forbidden region in classical mechanics becomes a region of reduced probability in quantum mechanics, not always a region of zero probability.

This is why tunneling is often described as a purely quantum effect. It has no true classical equivalent.

Important Examples

Tunneling is not just a mathematical curiosity. It is essential in many real physical systems.

In alpha decay, an alpha particle escapes from a nucleus by tunneling through the nuclear potential barrier. Classically, it would remain trapped because it does not have enough energy to climb over the barrier.

In scanning tunneling microscopy, electrons tunnel between a sharp metal tip and a surface. Because the tunneling current depends very strongly on distance, the device can detect extremely small surface features, even at the atomic scale.

In semiconductor devices, tunneling plays a role in tunnel diodes and other nanoscale electronic components. It also affects the behavior of electrons in very thin insulating layers.

In stars, tunneling helps nuclear fusion occur. Positively charged nuclei repel each other electrically, but tunneling allows them to get close enough for the strong nuclear force to act.

A Distance Sensitive Effect

One especially important feature of tunneling is its sensitivity to barrier width. If the transmission probability behaves like

$$
T \propto e^{-2\kappa a}
$$

then changing $a$ a little can change $T$ a lot.

For example, if all else stays the same and the width doubles, the exponent doubles, so the probability can become dramatically smaller. This is why tunneling often matters only over very short distances.

Rendering scientific sketch…

Exponential decay of the wave function in a barrier

Tunneling and Probability

The wave function itself is not directly the probability. The probability density is related to $|\psi|^2$. So if the wave function becomes smaller inside the barrier, the probability density becomes even smaller. Still, as long as it is not exactly zero, there remains some chance of finding the particle beyond the barrier.

This is the practical meaning of tunneling. Most particles may reflect, but some fraction transmit through the barrier.

Key Ideas to Remember

Quantum tunneling happens because the wave function extends into regions that are classically forbidden. In those regions, the wave function usually decreases exponentially instead of oscillating. A finite barrier can therefore be crossed with a nonzero probability, even when the particle energy is less than the barrier height. The transmission probability decreases rapidly for wider, higher, or more massive barriers.

Essential facts about quantum tunneling:
$$
E < V_0 \quad \text{does not imply} \quad T = 0
$$
Inside a forbidden region,
$$
\psi \text{ decays approximately as } e^{-\kappa x}
$$
with
$$
\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}
$$
and the transmission probability often behaves like
$$
T \propto e^{-2\kappa a}
$$

Quantum tunneling is one of the most important and surprising results of quantum mechanics, connecting the abstract wave function to real effects in nuclei, electronics, and microscopy.

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